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| Mirrors > Home > ILE Home > Th. List > enpr2d | GIF version | ||
| Description: A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| Ref | Expression |
|---|---|
| enpr2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| enpr2d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
| enpr2d.3 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| enpr2d | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | enpr2d.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
| 2 | ensn1g 6971 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
| 3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o) |
| 4 | enpr2d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
| 5 | 1on 6589 | . . . . 5 ⊢ 1o ∈ On | |
| 6 | en2sn 6988 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o}) | |
| 7 | 4, 5, 6 | sylancl 413 | . . . 4 ⊢ (𝜑 → {𝐵} ≈ {1o}) |
| 8 | enpr2d.3 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
| 9 | 8 | neqned 2409 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| 10 | disjsn2 3732 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
| 11 | 9, 10 | syl 14 | . . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
| 12 | 5 | onirri 4641 | . . . . . 6 ⊢ ¬ 1o ∈ 1o |
| 13 | 12 | a1i 9 | . . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o) |
| 14 | disjsn 3731 | . . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o) | |
| 15 | 13, 14 | sylibr 134 | . . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅) |
| 16 | unen 6991 | . . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) | |
| 17 | 3, 7, 11, 15, 16 | syl22anc 1274 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) |
| 18 | df-pr 3676 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 19 | df-suc 4468 | . . 3 ⊢ suc 1o = (1o ∪ {1o}) | |
| 20 | 17, 18, 19 | 3brtr4g 4122 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o) |
| 21 | df-2o 6583 | . 2 ⊢ 2o = suc 1o | |
| 22 | 20, 21 | breqtrrdi 4130 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1397 ∈ wcel 2202 ≠ wne 2402 ∪ cun 3198 ∩ cin 3199 ∅c0 3494 {csn 3669 {cpr 3670 class class class wbr 4088 Oncon0 4460 suc csuc 4462 1oc1o 6575 2oc2o 6576 ≈ cen 6907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-v 2804 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-1o 6582 df-2o 6583 df-er 6702 df-en 6910 |
| This theorem is referenced by: isnzr2 14201 |
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