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| Mirrors > Home > MPE Home > Th. List > Mathboxes > unir1regs | Structured version Visualization version GIF version | ||
| Description: The cumulative hierarchy of sets covers the universe. This version of unir1 9729 replaces setind 9660 with setindregs 35267. (Contributed by BTernaryTau, 30-Dec-2025.) |
| Ref | Expression |
|---|---|
| unir1regs | ⊢ ∪ (𝑅1 “ On) = V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | setindregs 35267 | . 2 ⊢ (∀𝑥(𝑥 ⊆ ∪ (𝑅1 “ On) → 𝑥 ∈ ∪ (𝑅1 “ On)) → ∪ (𝑅1 “ On) = V) | |
| 2 | vex 3445 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | r1elss 9722 | . . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) |
| 4 | 3 | biimpri 228 | . 2 ⊢ (𝑥 ⊆ ∪ (𝑅1 “ On) → 𝑥 ∈ ∪ (𝑅1 “ On)) |
| 5 | 1, 4 | mpg 1799 | 1 ⊢ ∪ (𝑅1 “ On) = V |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3441 ⊆ wss 3902 ∪ cuni 4864 “ cima 5628 Oncon0 6318 𝑅1cr1 9678 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-regs 35263 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7363 df-om 7811 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-r1 9680 |
| This theorem is referenced by: fineqvr1ombregs 35275 |
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