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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 6775 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o) |
4 | enpr2d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
5 | 1on 6402 | . . . . 5 ⊢ 1o ∈ On | |
6 | en2sn 6791 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o}) | |
7 | 4, 5, 6 | sylancl 411 | . . . 4 ⊢ (𝜑 → {𝐵} ≈ {1o}) |
8 | enpr2d.3 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
9 | 8 | neqned 2347 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | disjsn2 3646 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
11 | 9, 10 | syl 14 | . . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
12 | 5 | onirri 4527 | . . . . . 6 ⊢ ¬ 1o ∈ 1o |
13 | 12 | a1i 9 | . . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o) |
14 | disjsn 3645 | . . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o) | |
15 | 13, 14 | sylibr 133 | . . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅) |
16 | unen 6794 | . . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) | |
17 | 3, 7, 11, 15, 16 | syl22anc 1234 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) |
18 | df-pr 3590 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
19 | df-suc 4356 | . . 3 ⊢ suc 1o = (1o ∪ {1o}) | |
20 | 17, 18, 19 | 3brtr4g 4023 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o) |
21 | df-2o 6396 | . 2 ⊢ 2o = suc 1o | |
22 | 20, 21 | breqtrrdi 4031 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∪ cun 3119 ∩ cin 3120 ∅c0 3414 {csn 3583 {cpr 3584 class class class wbr 3989 Oncon0 4348 suc csuc 4350 1oc1o 6388 2oc2o 6389 ≈ cen 6716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 |
This theorem is referenced by: (None) |
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