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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 6790 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o) |
4 | enpr2d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
5 | 1on 6417 | . . . . 5 ⊢ 1o ∈ On | |
6 | en2sn 6806 | . . . . 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 2354 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | disjsn2 3654 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
11 | 9, 10 | syl 14 | . . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
12 | 5 | onirri 4538 | . . . . . 6 ⊢ ¬ 1o ∈ 1o |
13 | 12 | a1i 9 | . . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o) |
14 | disjsn 3653 | . . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o) | |
15 | 13, 14 | sylibr 134 | . . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅) |
16 | unen 6809 | . . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) | |
17 | 3, 7, 11, 15, 16 | syl22anc 1239 | . . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) |
18 | df-pr 3598 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
19 | df-suc 4367 | . . 3 ⊢ suc 1o = (1o ∪ {1o}) | |
20 | 17, 18, 19 | 3brtr4g 4034 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o) |
21 | df-2o 6411 | . 2 ⊢ 2o = suc 1o | |
22 | 20, 21 | breqtrrdi 4042 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∪ cun 3127 ∩ cin 3128 ∅c0 3422 {csn 3591 {cpr 3592 class class class wbr 4000 Oncon0 4359 suc csuc 4361 1oc1o 6403 2oc2o 6404 ≈ cen 6731 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-1o 6410 df-2o 6411 df-er 6528 df-en 6734 |
This theorem is referenced by: (None) |
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